Currently mages only get two weapon skills. 1h bludgeon and 1h piercing. That being the case, you can't earn skills points in skills you don't have. So the answer to both questions is no.

I would like to elaborate and say that weapon skills are not that significant to a mage anyhow. Sure, you might use them to get some exp while you are first starting out at lower levels, but weapon skill points for melee damage at lower levels does not have a significant effect on the end results. Neither does having the skill at all for the weapon in question (that I have noticed) at lower levels.

If from a little testing, your sword does more damage than any staff or dagger you can get your hands on, use it. You will quickly become very much more dependant on your magic for damage anyways.

Most of the time classes cannot use weapons that they never gain skills in (for example, there are very few 2h slashing weapons that can be wielded by rogues). The few exceptions that do exist are generally less powerful weapons. Of course, mages are at the bottom of the proverbial heap in terms of weapon choices, so even a sub-par sword might be better than your average dagger or quarterstaff.